Evaluate the following logarithm

Question

Evaluate:log_e 2e³ + 2log_e 3 - log_e 18

Answer 1

Properties of Logarithms:

log_ex = ln x 

ln e = 1

 

a) log_x(ab)= log_x(a)+ log_x(b)

b) log_x(a/b)= log_x(a) - log_x(b)

c) log_x(aⁿ) = n * log_x (a)

 

so... log_e 2e³ = (ln 2) + (3 * ln e) (properties a and c)

        2log_e3= ln (3²) = ln 9 (reverse of property c)

        

       ln 2 + 3*ln e + ln 9 = 3* ln (9*2*e) (combine the two and use property a )

       

       3*ln 18e - ln 18 =3 ln (18e/18) = 3 ln e  (combine with 3rd eq, use property b and simplify)

      since ln e = 1, 3 ln e = 3

 

 final answer = 3

Answer 2

ln(2e^3)+2ln(3)-ln(18)

ln(2)+ln(e^3)+2ln(3)-ln(18)

0.693147+3+2*(1.098612)-2.890371 (from tables)

0.693147+3+2.197224-2.890371

3.693147+2.197224-2.890371

5.890371-2.890371

3

or ln(2)+ln(e^3)+ln(3)+ln(3)-ln(18)

    ln(2)+ln(3)+ln(3)-ln(18)+ln(e^3)

    ln(2*3*3)-ln(18)+ln(e^3)

    ln(18)-ln(18)+ln(e^3)

    ln(18/18)+ln(e^3)

    ln(1)+ln(e^3)

    0+ln(e^3)

    ln(e^3)

    3